Finding x in cos(x)=exp(-2x): Analytical Solutions?

[Swest]

Hello all

Can anybody see a way to analytically find x in the expression:

cos(x) = exp(-2x)

By inspection x=0 is obvious, and numerically we find x=1.5232 is also a solution, but is there a way to find these values by rearranging the above expression? It's one of those that looks simple but isn't, any ideas?

Many thanks

 

[HallsofIvy]

That is not "simple" algebra. What you are attempting to do is solve an equation involving two different transcendental functions. In general there is no simple way to do that.

 

[Swest]

Thanks, I suspected as much. Always nice to be reassured that one isn't missing something obvious.

 

[mathwonk]

graphing them is instructive as it shows the graphs meet infinitely often, at points that become closer and closer to the zeroes of cos(x),

i.e. the solutions approach closer and closer to the integral multiples of pi/2.

 

[ice109]

graphing them is instructive as it shows the graphs meet infinitely often, at points that become closer and closer to the zeroes of cos(x),

i.e. the solutions approach closer and closer to the integral multiples of pi/2.

that make sense seeing as for large x the exponential is vanishing.

 

[Swest]

Yes, I did a quick MATLAB plot of the two and saw the trend that you describe, which again can be surmised from inspection, indeed the 2nd solution is very close to pi/2.